Unit · year 2
MU-202 · Linear Algebra II
Threads structure25 lectures6 theorems
Abstract vector spaces, operators, and the canonical forms that classify them.
PREREQUISITES
Lectures
| L01 | Vector Spaces over an Arbitrary Field — |
| L02 | The Steinitz Exchange Lemma |
| L03 | Invariance of Dimension |
| L04 | Quotient Spaces and the Dimension Formula |
| L05 | Dual Spaces and Dual Bases — |
| L06 | Bilinear and Sesquilinear Forms — |
| L07 | Inner Product Spaces and Adjoints — |
| L08 | Orthogonal Projection |
| L09 | Least Squares and the Normal Equations |
| L10 | Eigenvalues and the Characteristic Polynomial — |
| L11 | Criteria for Diagonalisability — |
| L12 | The Cayley–Hamilton Theorem |
| L13 | Minimal Polynomials |
| L14 | Invariant Subspaces and Triangularisation — |
| L15 | Generalised Eigenspaces |
| L16 | The Jordan Normal Form |
| L17 | Computing the Jordan Form and Matrix Functions |
| L18 | Self-Adjoint and Normal Operators |
| L19 | The Spectral Theorem |
| L20 | Quadratic Forms and Sylvester's Law of Inertia — |
| L21 | Positive-Definite Operators |
| L22 | The Singular Value Decomposition |
| L23 | Low-Rank Approximation and the Pseudoinverse |
| L24 | Tensor Products: A First Look — |
| L25 | Synthesis: Canonical Forms and What They Reveal |
Theorems in this unit
T-046
Invariance of dimension
Every basis of a vector space has the same cardinality.
T-047
The spectral theorem
A self-adjoint operator has an orthonormal basis of eigenvectors.
T-048
The Cayley–Hamilton theorem
Every matrix satisfies its own characteristic polynomial.
T-049
The Jordan normal form
Every operator over C is similar to a direct sum of Jordan blocks.
T-050
The singular value decomposition
Any matrix factors as a rotation, a scaling, and a rotation.
T-051
The orthogonal projection theorem
Best approximation in an inner product space is orthogonal projection.